A FIRST COURSE IN INTEGRAL EQUATIONS

A FIRST COURSE IN INTEGRAL EQUATIONS

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A FIRST COURSE IN INTEGRAL EQUATIONS Abdul-M ajid Wazwaz Saint Xavier University, USA lib World Scientific 1M^ Singapore New Jersey London Hong Kong

Published by World Scientific Publishing Co. Pte. Ltd. P O Box 128, Farrer Road, Singapore 912805 USA office: Suite IB, 1060 Main Street, River Edge, NJ 07661 UK office: 57 Shelton Street, Covent Garden, London WC2H 9HE Library of Congress Cataloging-in-Publication Data Wazwaz, Abdul-Majid. A first course in integral equations / Abdul-Majid Wazwaz. p. cm. Includes bibliographical references and index. ISBN 9810231016 1. Integral equations. I. Title. QA431.W36 1997 515\45--dc21 97-3749 CIP British Library Cataloguing-in-Publication Data A catalogue record for this book is available from the British Library. Copyright 1997 by World Scientific Publishing Co. Pte. Ltd. All rights reserved. This book, or parts thereof, may not be reproduced in any form or by any means, electronic or mechanical, including photocopying, recording or any information storage and retrieval system now known or to be invented, without written permission from the Publisher. For photocopying of material in this volume, please pay a copying fee through the Copyright Clearance Center, Inc., 222 Rosewood Drive, Danvers, MA 01923, USA. In this case permission to photocopy is not required from the publisher. This book is printed on acid-free paper. Printed in Singapore by UtoPrint

Contents 1 Introductory Concepts 1 1.1 Definitions 1 1.2 Classification of Linear Integral Equations 3 1.2.1 Fredholm Linear Integral Equations: 3 1.2.2 Volterra Linear Integral Equations: 4 1.2.3 Integro-Differential Equations: 6 1.2.4 Singular Integral Equations: 7 1.3 Solution of an Integral Equation 11 1.4 Converting Volterra Equation to ODE 15 1.5 Converting IVP to Volterra Equation 20 1.6 Converting BVP to Fredholm Equation 26 2 Fredholm Integral Equations 31 2.1 Introduction 31 2.2 The Decomposition Method 33 2.2.1 The Modified Decomposition Method: 38 2.3 The Direct Computation Method 43 2.4 The Successive Approximations Method 48 2.5 The Method of Successive Substitutions 52 2.6 Comparison between Alternative Methods 56 2.7 Homogeneous Fredholm Equations 59 3 Volterra Integral Equations 67 3.1 Introduction 67 3.2 The Adomian Decomposition Method 68 3.2.1 The Modified Decomposition Method: 73 3.3 The Series Solution Method 77 v

vi CONTENTS 3.4 Converting Volterra Equation to IVP 82 3.5 Successive Approximations Method 86 3.6 The Method of Successive Substitutions 91 3.7 Comparison between Alternative Methods 95 3.8 Volterra Equations of the First Kind 99 4 Integra-Differential Equations 103 41 Introduction 103 42 Fredholm Integro-Differential Equations 105 42.1 The Direct Computation Method: 105 42.2 The Adomian Decomposition Method: 109 42.3 Converting to Fredholm Integral Equations:... 118 43 Volterra Integro-Differential Equations 121 43.1 The Series Solution Method: 121 43.2 The Decomposition Method: 126 43.3 Converting to Volterra Integral Equation:... 131 43.4 Converting to Initial Value Problems: 134 5 Singular Integral Equations 139 5.1 Definitions 139 5.2 Abel's Problem 141 5.2.1 The Generalized Abel's Integral Equation... 146 5.3 The Weakly-Singular Volterra Equations 150 6 Nonlinear Integral Equations 157 6.1 Definitions 157 6.2 Nonlinear Fredholm Integral Equations 158 6.2.1 The Direct Computation Method. 159 6.2.2 The Decomposition Method 163 6.3 Nonlinear Volterra Integral Equations 173 6.3.1 The Series Solution Method 173 6.3.2 The Decomposition Method 177 A Table of Integrals 183 B Integrals of Irrational Functions 187 C Series Representations 189

CONTENTS vii D The Error and Gamma Functions 191 Answers To Exercises 193 Bibliography 205 Index 207

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Preface Mathematics, science and engineering students, both advanced undergraduate and beginning graduate, need an integral equations textbook that simply and easily introduces the material. They also need a textbook that embarks upon their already acquired knowledge of regular integral calculus and ordinary differential equations. Because of these needs, this textbook was created. From many years of teaching, I have found that the available treatments of the subject are abstract. Moreover, most of them are based on comprehensive theories such as topological methods of functional analysis, Lebesgue integrals and Green functions. Such methods of introduction are not easily accessible to those who have not yet had a background in advanced mathematical concepts. This book is especially designed for those who wish to understand integral equations without having the extensive mathematical background. In this fashion, this text leaves out abstract methods, comprehensive methods and advanced mathematical topics. From my experience in teaching and in guiding related senior seminar projects for advanced undergraduate students, I have found that the material can indeed be taught in an accessible manner. Students have shown both a lot of motivation and cabability to grasp the subject once the abstract theories were ommitted. In my approach to teaching integral equations, I focus on easily applicable techniques and I don't emphasize such abstract methods as existence, uniqueness, convergence and Green functions. I have translated my means of introducing and fully teaching this subject into this text so that the intended user can take full advantage of the easily presented and explained material. I have also introduced and made full use of some recent developments in this field. ix

X Preface The book consists of six chapters, each being divided into sections. In each chapter the equations are numbered consecutively and distinctly from other chapters. Several examples are introduced in each section, and a large number of exercises, with varying degrees of difficulty but being consistent with the material, are included to give the students constructive insights about the material and to provide them with useful practice. In this text, we were mainly concerned with linear integral equations, mostly of the second kind. Chapter 1 introduces classifictions of integral equations and necessary techniques to convert differential equations to integral equations or vice versa. Chapter 2 deals with linear Fredholm integral equations and the reliable techniques, supported by the new developments, to handle this style of equations. In Chapter 3 the linear Volterra integral equations are handled, using the recent developed techniques beside standard ones. In Chapter 4 the topic of integro-differenfcial equations has been handled and reliable techniques were implemented to handle the essential link between differential and integral operators. Chapter 5 introduces the treatment of the singular and the weakly singular Volterra type integral equations. Chapter 6 deals with the nonlinear integral equations. This topic is difficult to study. However, recent developments have shown improvements over existing techniques and allow this topic to be far more easily accessible for specific cases. A large number of nonlinear integral examples and exercises are investigated. Throughout the text, examples are provided to clearly and throughly introduce the new material in a clear and absorbable fashion. Many exercises are provided to give the new learner a chance to bulid his confidence, ease and skill with the newly learned material. The text has four useful Appendices. These Appendices provide the user with the integral forms, Maclaurin series and other related materials which are needed to be used in the exercises. Finally, this book is suitable for a one-semester course in integral equations, hoping it will be useful for anyone intersted in integral equations. I am indebted to my wife, my son and my daughters who provided me with their encouragement, patience and support. I am also indebted to Professor Louis Pennisi who first introduced

Preface xi me to integral equations and instilled in me great love for it. His excellent methods of teaching have inspired me greatly. The author would highly appreciate any note concerning any error found and for any constructive suggestion. Chicago, IL 1996 Abdul-Majid Wazwaz